280 HISTORY OF THE THEORY OF NUMBERS. [CHAP, x
Material on (1) will be given in the chapter on partitions in Vol. II. J. H. Lambert,7 by expanding the terms by simple division, obtained
in which the coefficient of xn is r(ri). Similarly, he obtained (4) from the left member of (3) .
E. Waring8 reproduced Euler's6 proof of (2).
E. Waring9 employed the identity
n (xk-i)^xb-xb-l-xb-2+xb-5+xb-7- - . . . =A,
*=i
the coefficient of xb~v; for v^n, being (—1)' if v-(3z*=*=z)/2 and zero if v is not of that form. If ra^n, the sum of the mth powers of the roots of A = 0 is cr(m). Thus (2) follows from Newton's identities between the coefficients and sums of powers of the roots. He deduced
where c= =±= 1 or 0 is the coefficient of xb~m in series A. Let
where p ranges over the primes 1, 2, 3, 5, . . ., n. If m^n, the sum of the mth powers of the roots of A' = 0 equals the sum <rf(m) of the prime divisors of m. Thus
We obtain (5) with <r replaced by cr', and c by the coefficient of xb'~~m in series Af. Consider
n (s*-l) =xb-xb-l-xb~2l+xb-5l+ . . . =B,
y-l
with coefficients as in series A. The sum of the (Zm)th powers of the roots of B = 0 equals the sum cr(0 (m) of those divisors of m which are multiples of I. Thus
with the same laws as (2). The sum of those divisors of m which are divisible
7Anlage zur Architectonic, oder Theorie des Ersten imd des Einfachen in der phil. und math.
Erkenntniss, Riga, 1771, 507. Quoted by Glaisher.75 8Meditationes Algebraicse, ed. 3, 1782, 345. 8Phil. Trans. Roy. Soc. London, 78, 1788, 388-394,